Skip to main content

Full text of "A hybrid computer technique for measuring human describing functions and remnant in closed-loop tracking tasks."

See other formats


' ' 



A HYBRID COMPUTER TECHNIQUE FOR 

MEASURING HUMAN DESCRIBING FUNCTIONS AND 

REMNANT IN CLOSED-LOOP TRACKING TASKS 



Roy Dale Warren 




NAVAL POSTGRADUATE SCHOOL 

Monterey, 




1 H el^d 3 o 



A HYBRID COMPUTER TECHNIQUE 
FOR 
MEASURING HUMAN DESCRIBING FUNCTIONS AND 
REMNANT IN CLOSED-LOOP TRACKING TASKS 

by 

Roy Dale Warren 



Thesis Advisor 



Ronald A. Hess 



June 1972 



Approved for public release; distribution unlimited. 



A Hybrid Computer Technique: 
for 
Measuring Human Describing Functions and' 
Remnant in Closed-Loop Tracking Tasks 



by 



Roy Dale Warren 
Lieutenant Commander, United States; Navy 
B.S., United States Naval Academy,. L963 



Submitted in partial, fulfillment of the 
requirements for the degree of" 



AERONAUTICAL ENGINEER 

from the 

NAVAL POSTGRADUATE SCHOOL 
June 1972 



ABSTRACT 

The measurement of the human describing function and 
remnant in a compensatory tracking task, is undertaken.. 
These measurements are obtained through, the application-, of 
the fast Fourier transform technique on a hybrid (analog-— 
digital) computer. This method processes" the. data:, in. reai 
time with minimal core storage and the results: are aval]. "able 
immediately upon completion of the tracking run.. 



TABLE OF CONTENTS 

I. INTRODUCTION ' SE 

A. BACKGROUND 99 

B. COMPENSATORY TASKS AND QUASI-LINEAR NATION. LO". 

II. SPECTRAL ANALYSIS 121 

A. PERIODIC SIGNALS 122 

B. TRANSIENT SIGNALS 14- 

C. RANDOM SIGNALS 336c 

III. DESCRIBING FUNCTION AND REMNANT RELATIONS 18 

A. FREQUENCY DOMAIN RELATIONS 18 

B. FINITE RUN LENGTH 19' 

C. DESCRIBING FUNCTION AND REMNANT 21 

D. SINUSOIDAL INPUTS ZZ 

IV. COMPUTER MECHANIZATION 26 

A. EXPERIMENTAL SET-UP 26 

B. FAST FOURIER TRANSFORM •■ 26 

V. RESULTS 30 

COMPUTER PROGRAM 52 

LIST OF REFERENCES 63 

INITIAL DISTRIBUTION LIST 64 



FIGURES 

1. HUMAN OPERATOR IN A COMPENSATORS: TRACKING TASK'. 3j4. 

2. QUASI-LINEAR MODEL OF HUMAN OPERATOR. IN- A. 
COMPENSATORY TRACKING TASK 35: 

3. MEASUREMENT OF KNOWN |-=— J ELEMENT 3 6 

s 

4. MEASUREMENT OF KNOWN ( ^-pf ELEMENT 37 

5. MEASUREMENT OF KNOWN =r ELEMENT 33 

cs + n 

7 

6. MEASUREMENT OF KNOWN {-= ) ELEMENT 39 

s + 2s + 2 

7. AMPLITUDE OF HUMAN DESCRIBING FUNCTION. FOR 

Y (s) = 1.0 (SUBJECT 1) 40. 

8. PHASE OF HUMAN DESCRIBING FUNCTION FOR Y (s) = ILIX 

(SUBJECT 1) 41" 

9. REMNANT OF HUMAN DESCRIBING FUNCTION FOR 

Y (s) =1.0 (SUBJECT 1) ; 42 

c 

10. AMPLITUDE OF HUMAN DESCRIBING FUNCTION FOR 

Y (s) =1.0 (SUBJECT 2) 43 

11. PHASE OF HUMAN DESCRIBING FUNCTION FOR 

Y (s) = 1.0 (SUBJECT 2) 44 

12. REMNANT OF HUMAN DESCRIBING FUNCTION FOR 

Y c (s) = 1.0 (SUBJECT 2) 45 

13. AMPLITUDE OF HUMAN DESCRIBING FUNCTION FOR 

Y (s) = (-^-) (SUBJECT 1 ) ■ 46; 

14. PHASE OF HUMAN DESCRIBING FUNCTION FOR 

Y (s) = (-1—) (SUBJECT 1 ) 47 

C- s 



15. REMNANT OF HUMAN DESCRIBING FUNCTION FOR 

Y (s) = (-i-) (SUBJECT!) 48; 

16. AMPLITUDE OF HUMAN DESCRIBING FUNCTION' FOR 

Y (s) = (-^—) (SUBJECT 2) 4-9: 

17. PHASE OF HUMAN DESCRIBING FUNCTION FOR 

Y (s) = (-^-) (SUBJECT 2) 5D_ 

18. REMNANT OF HUMAN DESCRIBING FUNCTION FOR 

Y (s) = (-^— ) (SUBJECT 2) 511 

c s 



TABLE OF SYMBOLS 



A f REAL PART OF FOURIER TRANSFORM OF f (t). AT EEEQUENCY 



k 



k 



w k 



B f IMAGINARY PART OF FOURIER TRANSFORM OF E(t) ATT 



FREQUENCY, u 



C (t) OUTPUT SIGNAL 

e(t) ERROR SIGNAL 

f(t) ARBITRARY SIGNAL 

F(jm) FOURIER TRANSFORM OF f (t) 

F(ju)) COMPLEX CONJUGATE OF F(jw) 

F(n) FOURIER COEFFICIENT FOR A PERIODIC SIGNAL f (t) 

F(n) COMPLEX CONJUGATE OF F (n) 

H(s) SYSTEM TRANSFER FUNCTION 

i (t) INPUT SIGNAL 

i (t) INPUT SIGNAL OF FINITE DURATION 

n(t) REMNANT SIGNAL 

p(t) TOTAL OPERATOR RESPONSE SIGNAL, p'(t) + n(t) 

p f (t) LINEAR OPERATOR RESPONSE SIGNAL 

T PERIOD OF TOTAL RUN LENGTH 

T, PERIOD FOR FREQUENCY, oj, 

Y (jco) CONTROLLED ELEMENT TRANSFER FUNCTION 

Y (ju>) HUMAN DESCRIBING FUNCTION 
P 

<J> (t) autocorrelation function OF f (t) 

ff 

4> (t)CROSSCORRELATION FUNCTION OF f (t) and" f ? (t) 

$ (n) POWER SPECTRAL DENSITY OF PERIODIC SIGNAL, f (t) 



$ (n) CROSS- POWER SPECTRAL. DENSITY OF" EERXODIC: SIGNALS 
f l f 2 

f 1 (t) and f 2 (t) 

$ (w) POWER (ENERGY) SPECTRAL. DENSITY" OF" tl{t) 

# (a)) CROSS- POWER (ENERGY) SPECTRAL DENSITY" OF" f i. .(t) and' 
£ l t 2 

f 2 (t) 

0), FREQUENCIES OTHER THAN THOSK IN THE INPUT- SINUSOIDS. 

Uh FREQUENCIES OF INPUT SINUSOIDS 



ACKTJOWLED GEMENTS 

This thesis was written with, the dose support- ancL 
never ending assistance of Asst.. Prafessor R.. A.. Hess:. The, 
contributions of Asst. Professor M.'- H.. Redlin. as a' second- 
reader on this thesis are hereby acknowledged'.. In. addition, 
the contributions of Mr. Robert E.. Limes- and" Mr.. Albert . 
Wong of the Computer Laboratory staff are acknowledged" fbr/ 
their assistance in the computer programming;.. 



I. INTRODUCTION 

A . BACKGROUND 

Many of the important tasks performed try pilots" are:- akin 
to those performed by linear servomechanisms .. In. situations 
such as this, the pilot can be modeled by a set- ofr" constant- 
coefficient linear differential equations:.. Ih\ the frequency 
domain, such a model is often referred to as a "human, pilot:: 
describing function," The term "describing function" is 
preferred to "transfer function" to emphasize the fact that 
this pilot model is approximating a nonlinear element, and.'. 
is valid only for the particular inputs,, system dynamics- and 
task at hand. 

The pilot-describing function is useful in studying two 
classes of problems. First, the describing functions 
measured in the piloted simulation of a given, aircraft 
and task can be utilized in the subsequent stability and 
control analysis of this aircraft. Once the pilot's 
describing function for a particular task has been measured, 
he can be analytically replaced by his describing function 
in the analyses normally associated with the study of linear 
feedback systems. Second, actual measurement of pilot 
describing functions in ground simulation or in\ flight tests 
can be used to determine how a particular aircraft, or flight. 
task affects pilot behavior. Knowledge of pilot describing 
functions consequently provides valuable information for 



the aircraft designer. 

It is the problem of human describing function measure- 
ment which forms the basis of this research... Thes hybricL 
computer program which has resulted can be utilized in. 
virtually any study involving human behavior in compensatory 
tracking tasks. 

B. COMPENSATORY TASKS AND QTTASI-LINEAHIZATTQN 

The compensatory tracking task.,, shown in. Eigure 1, , 
assumes that the error signal is the only information that 
the operator is receiving. In this study the operator 
attempts to minimize a visual error signal by using a hand- 
operated controller. Tracking situations such as this: are 
often encountered in aircraft flight control;, e:.g..,. a pilot 
attempting to maintain some desired pitch attitude in. the 
presence of atmospheric turbulence.. It has been, shown that. 
in tasks such as this, the operator is nonlinear and" time 
variant in behavior. He may, however, be successfully 
modeled in a quasi-linear fashion [Ref. 1] .. This quasi- 
linearization implies that his response to visual stimula- 
tion is largely linear and time invariant; i.e., his 
dynamics are largely those of constant-coefficient linear 
differential equations. To account for nonlinear and/ or 
time-varying behavior, the model includes a remnant, signal 
as shown in Figure 2. The remnant is that portion. of: the 
operator's output which is not linearly correlated with the 
input. The human operator model thus consists of a linear 
describing function, Y ( jco) , determined from the quasi-linear 

10 



analysis, and a remnant, n(t). 

It should be noted that this quasi— linear: raodell is: of: 
little use if the remnant is relatively large, sincei then. 
the operator ' s behavior is predominantly nonlinear and/ on- 
time varying. 

To determine the human operator model it is: necessary 
to calculate the linear describing: function. Y_ ( jco) ,. and the 
remnant power spectral density, <6 .(to),, from physical- 
measurements of signals of finite duration in: a. laboratory 
experiment. The input signal must appear to the operator 
to be random, although it need not be truly random, and 
the operator must be well trained; i.e.. , not undergoing 
adaptation or learning [Ref. 21. 

In order to measure the human describing function. and 
remnant, one of three techniques can be employed.. These 
are the direct Fourier analysis of the system signals,, the 
use of crosscorrelation methods, and a model optimization 
technique. These methods are discussed in Ref.. 3c.. 

If, as done here, direct Fourier analysis or cross- 
correlation methods are employed in measuring the human 
describing function and remnant, then the concept of 
spectral analysis must be introduced. The next section is 
devoted to this topic. A more thorough treatment can: be 
found in Ref. 4, Appendix D. 



11 



II. SPECTRAL ANALYSIS 

A. PERIODIC SIGNALS 

A periodic signal, f(t), with a fundamental frequency 
to, and period T, satisfying the Dirichlet conditions [Kef.. 44,, 
p. 579] , may be represented by a Fourier series 



where 



f(t) = I VM***!* 
n =-oo 

(n) = -^^/^(tie-^l-dt.. 
1 J-T/2 



The autocorrelation function for the ahove periodic: 
signal is defined as 



* ff (T) - 



1_ /t/2. 

T J-T/2 



f (t)f (t+x)dt.. 



This can be written 



<f> ff (T) = 



and is equivalent to 



■+/ 



T/2 f(t) I F(n)e^ na) i (t+T) dt 
T/2 n=-°° 



* ff (T) - 



I F(n)e^i^(4_) /"^faje^^tdt 
n =-oo \ L I J - T /2 

With F(n) denoting the complex conjugate of F (n) , 
♦ ff (t) = I F(n)e^ naJ l T F(n) = \ F (n) F (n) e J naj L T .,, 



n=-oo 



n--a= 



n =-oo 



F(nJ 



2 jnoj.x 
e J J. 



12 



The power spectral density, <£ ff (n) '„ is defined 



•«w - -t-jgj-w^ 1 * * 



-T/2 
and it can be shown that 



$ ff (n) = 



2 



F(n) 
It can be seen from this relationship that 



*ff (T) = I tffWe 1 ™^ 
n=- cc 

The crosscorrelation function, ^-p -p (t)', of two: periodic 

12 
signals may be found in a similiar manner if both signals 

have equal fundamental frequencies, w , and bath, signals- 
satisfy the Dirichlet conditions^ Assuming these conditions 

are met, then 

T/2 

♦ f f (T) = -\-[ f x (t)f (t+T)dt .. 

f l f 2 T J-T/2 L 

In a fashion similar to that for single signals, it can be 
shown 

00 

*f f (T)= l F 1 (n)F 2 (n) e jnw l T 
1 2 n=-°° 

The cross-power spectral density, $_ ,_ (n) , is defined 

*! 2 



£ 



f f (n) = -=- / ' <J> (t) e J Ldr 

12 X ./-T/2 r L r 2 



and it can be shown that 



•fifj 011 = F l (n)F Z <n> 



Using this relationship 



1 2 n=-°° 1 2 



13 



B. TRANSIENT SIGNALS 

A signal, f (t) , is defined to be transient if: 

lim f (t) = tt.. 

If the transient signal r f (t) ,. satisfies the Dirxchlefc 
conditions in any finite interval, and if 






f(t) 



dt < 



then the signal may fee expressed as a Fourier integral-"- [Kef .5, 
p. 279]. Under these conditions, the Fouriex: integral", , 



f (tj = JL-f F(jco)e jajt du) 

J 1 — CO 



gives the values of f(t) at all points, including those where the 
function is not continuous. The Fourier transform of: f (t) is 

/•-CO 

FCjw) =/ f(t)e~ ja)t dt 



The autocorrelation function for - the nonperiodic signal 
is defined 

/*CO 

f(t)f(t+T)dt 



* ff (T) 



** — c 



This can also be written 



♦ ff (T] -j 



h: 



F(ja)) 



2 ioJT, 
e J dm 



Letting 



* ff Cw) = 



F(juj) 



// 



where $ __ (co) is defined to be the energy spectral density of 
the signal f(t), it can be shown that 



14 



<j> ff (T.)e IU)T dT 

-ex, 

Thus it can be seen that the energy . density spectrum and: 
the autocorrelation function of a transient signal! are_ bl 
Fourier transform pair, 

00 



J — oo 



and 



m/ — C 



$ ff Ctd) =/ <b ff {r)e ]a)T dr 



If two transient signals r f. (t) and f _ (t) ,, each satisfy 
the Dirichlet conditions in all finite intervals,, and if J 



//TOO 
f 1 (t) dt < « and / f 2 ( t) 
— oo 



dt « 



then 



^ oo /^oo 

*f f (T) = / f l (t)f 2 (t+T)dt = 2F"/ f " I (^)F z (jw)e ja)T duj 

12 •/ — °° J —CO 



Now with 

* f * (oj) = F (jw)F (jai) 
r l r 2 * Z 

where <a> , (oj) is defined as the cross -energy spec trail 

r l 2 
density of the signals f , (t) and f _ (t) , it can be shown; that 



# f f (») =/ * 

r l 2 y-oo r i 



, .-TUT, 

r: (x)e J dx 
2 



and has as its inverse transform 

$*■ r= (a))e- ia)T d'a) 
- f L f 2 

The transient signals f. (t) and f„(t) are said to be 

linearly uncorrelated when <J>_ - (t) = for all x. 

t l t 2 



15 



C. RANDOM SIGNALS 

In general, a random signal,, rT(.t) ' ,, from a. stationary,, 
ergodic random process [Ref. 4 r p Z7S] ,. does not have: ar 
Fourier transform since 






f (t) 



dt 



is not finite. An autocorrelation function may be defined 
for the random signal f(t) as 



*-£ 



♦ ff (t) = tIM ^-/ f (t)f(t+r)dt .. 

Since <{> ff: (T) satisfies the Dirichlet conditions f on: all 
finite intervals and 



wf — c 



* ff (T) 



dx < 



it can be represented by a Fourier integral.. It can then be 
shown that 



♦ff (T) " 2T-r*ff (u)eJUTd - 

J — CO 



where $_ f (w) is defined as the power spectral density of 
the signal f (t) . $ ff (u)) is the Fourier transform of (J) ff (x), 

• ff C«) =/ <f> ff (T)e" ja)T dx ., 

«/ — CO 

If there exist two random signals, f" (t) and f 2 (t) , 
which are sample functions from two different random processes, 
each of which are stationary, ergodic,. and jointly ergodic, 
then the crosscorrelation function: is given, as 

4>. 



f l f 2 



(x) = LIM -L-f f (t)f_(t+x)dt = JL-lj $_ _ (w)e jloT da3, 
T-^oo 2T J-T ± z ZTT J-co r l r 2 



16 



where $_ ^ (oj) is defined as the cxass-powex spectral' 



density of the random signals f T (t) and fL.(t).. <kp.fr ( T ) 



has: 
the Fourier transform 



-/ 



CO 



The 



random signals f, (t) and £_({t) are: said to: be: 



linearly uncorrelated if $._ - (x) = (I for" all. xt. 

r l 2 



17 



IH . DESCRIBING FUNCTION AND REMNANT RELATIONS 

A. FREQUENCY DOMAIN RELATIONS 

Again consider Figure 2 with, the input a sampfe: function- 
from an ergodic random process. It is 7 seen. that. 

E(jw) = ICjoiJ - C(jw) , 
where the Fourier transforms areas- defined, in Section B?. . Now 



C(jw) 



N(jw) +- Y (jcu)E(joo) 



Y- C (jw) ,. 



then 



E(ju)) 



I(jgj) - N(ja))Y c (joo) 



1 + Y (ja))Y (ju>) 
p c 



Finally, after multiplying by I(jco)',. the complex conjugate 
of the input, 



I(ja))E(jw) = 



I(ju)) I(ju>) - N(jo3)Y^(joo) 

c 



1 + I (ju)Y c (i(o) 



In a similar manner, 



P(joi) = Y (joi)E(joj) + N(joo) 

ir 

and then substituting for E(jcu) from above, 



P(jw) = 



Y (jw)l(ju) - N(joj)Y (ju))Y (jw) 
F P ^- 



1 + Y p (jaOY c (jco) 



+ N(jw) 



and 



Kjco)P(jw) = 



Kjw) 



Y (ju))I(joo) + N-(ju>) 



1 + Y (ia))Y (jai) 
p a c j. 



Likewise, T( jw)C ( jcj) , E( jca) P ( jco) , E( jta) E ( jaj) , P ( ±ui) P ( jw) , 
and c (jco)C(jto) may be calculated. The results., to be. 
utilized shortly, are shown in Table L.. 



18 



B. FINITE RUN LENGTH 

Since it is impossible to have experimental runs' of" 
infinite duration „ measurements using finite time- histories - 
are necessary. Reference 6" indicates that finite.- ran. 
lengths can be handled analytically" as: follows:: Ifi i^t) 
is the input and defined 

(ii(t) ,, - T <_ tt <fc T 



^(t) 



a „ ELSEWHERE 



then this function can be considered to be transient: and; 
have a Fourier transform 

iCjco) =J i (t)eT i(Ut at .. 

— oo 

The other system signals and their transforms can be- defined 
in precisely the same manner.. If the. run time, T~, . is large; 
enough to ensure accurate power spectral, measurements , yet. 
finite so that the respective Fourier - transforms exist, 
then the following spectral relations are valid [Ref.. 6] . 



and 



$ . - ( CO ) = - m 



§ - (co) = m 
rp v ' T+°° 



2T 



Kjoi) 



2T 



KJCO)P(JU)) 



where $. . (co) and § . (co) are power and cross-power' spectral 
densities respectively. Utilizing Table I, 



» , , LIM j 1 
*ip< w > = T— | 2T 



;i(j(d)I(ju>)Y_(ju>) +- I ( jco) N ( jco) Y c ( jtu) 
[ 1 + Y e: (jco)Y c Jj.co) 



but 



LIM 



I(jco) I (jco) 



2T 



= $". . (co) 

n 



19 



and 



thus 



LIM 



2^- I(juON(iw) 



= $-. (oj) 

in 



*ip (a)) = 1 + Y (jco)Y (jco) 



Now, 



/°° _ . 
<|).. (to) e " It0T dT. 
T in 



Since by definition the remnant,. n(t),. is- linearly y uncorr elated 



with the input, i (t) „ then $ . (t) = Q for all t... Thus 



$ . (co) = and 
in 



3F ha) $ . . (co) 
* / \ P ii 

1P * l+r p (ja))Y c (ja)) 



In a like manner. 



$ . (co) = m 



^- I(jco)E(}co) 



or 



*. (to) = 
ie 



$.. . (co) - $ . (to) Y (jco) 
ii in. c J 

1 + Y" (jco)Y (jco) 
p j c j 



Again since n(t) and i(t) are linearly uncorrelated, 



<j> in (T) = for all t, then $. (to) = 0.. Thus 



$. (co) = 
ie 



a. . (co) 
ii 



1+Y (jco)Y (jco) 
p J c 



Also 



$ (co) = 
pp v ' T-»-« 



^- P(jto)P(jto) 



$ (0)) 

pp 



LIM ( 1 F 1 ^)*" (jco)+N-(jto) 



T+°° J2T 



Ll + Y p (jco)Y c (jco) 



r(J3Q)Y p (jco)+N(jco) 



Ll + Y p (jco)Y c (jco) 



20 



or 



$ (co) 
PP 



*..(&)) I Y (jco) I +$ . (co)Y (jco)+$. (co)Y (jco)-K> (co) 
11 I p J 1 ex p a m p nn 



p c • 



Again, since $. (co) 
3 ' in 



> n .(.) = Q„ 



$ (to) 
PP 



$.. (co) 
11 



Y CjCD) 
Jr 



'+$ (co) 
nn 



l+Y p (jo))Y c (jco) 



2 



C. DESCRIBING FUNCTION AND REMNANT 

From $ . (co) , $. (to), and $ (co) , Y' (jco) and: $: (co) may 
ip le pp p J ■ nn. 2 - 

be found. Utilizing $. (co) and i. (co) 

le in v ' ' 



IP 

$..(03) = $.. (co) fl+Y (jco)Y (jco) 
li re I p J c. 



and 



♦ „( U ) = , ^ v *ip("> p--w p «B»J^ c ftfrO| • 



11 



Y p (jco) 



Thus, 



* ie (») [l+Y p (jo.)Y c (j U )] = T --y r « ip(u ) [l+YpOlY^fju.)] , 



or 



« +Vr foO 

Y (jco) = . ip , , 
P $ ie (w) 



Also, from $ (co) . 
PP 



$ (co) = 
nn 



l+Y p (jco)Y c (jco) 



$ (co)- 

PP 



Y (jco) 
P 



1JL 



(CO) . 



In addition, Y c (jco) can be determined and calculated from 



$ ic (co) = $ in (co)Y c (jco) + $. a (co)Y^(jto)Y_(jio)' 



le 



Again $ in (w) = 0, thus 



21 



«. Cui) = $. (oo)Y (jco)Y (jw) 
re re p c. 

But 

$.. (oj) Y (too) = $ . (to) , . 
re p a ip 

thus 

$ . (to) 

Y ](i) = -r t r- 

rp 
The functions in Table II" Form the.- basis: of f thee: describing 
function measurement technique utilized in this - study-." . 

D. SINUSOIDAL INPUTS 

The relations in Table II. are predicated on the existence 

of a random input. In experimental work, a" random appearing 

input is often used and can be generated as a summation of 

sine waves [Ref. 31. The input can thus be represented by 

n 
iftl = ]£ A, sinor.t: 
k=l * k 

where the or are chosen such that in a finite, run: length 

there will exist an integral number of periods or: complete 

cycles. In this analysis a run time of 150 seconds was 

used and 0.08 < or <_ 40. Q radians per second. 

Utilization of a sinusoidal input results in system 

signals that have both random and periodic components. With 

what is now a mixed signal, the question arises as to which 

power spectral relationship should be used.. The solution is 

to use the periodic power spectral relationships, for: measurer 

ments made at the input frequencies and to use the. random, 

finite relationships at all other frequencies. 



22 



It should be noted that if the: experimental! rnir length , 
T, is large and contains an integral niMber of: periods: of - 
each of the input sinusoids,, then the. Fourier transforms' of : 
the periodic and finite random signals differ- only by a: 
constant of proportionality [Ref.. 61". • 

In the periodic case, it was shown: that- 

F k (n) = ^-f^ftUe^^at „ 
and for the random signal , 

/•CO 

J — CO 

If it is recalled that f (t) - far- t<-T and. t>T y , and: 
letting T = m k T kr where m, is the number of periods, T, , of 
frequency to, , then 



f(t)e L dt = 2-Tj- J f(t)e 1 dt. , 

T. k k J -ra, T n 



F k (n) = A 

^k - ~*- iu k^k 



or 



F fc (n) = Jjr-j f (t)e ^k^dt ; 



thus it can be seen that 

2TF ]c (n.) = F(jco k ) .. 

It should be noted that the same expansion. of: the: limits 
on the integral can be used for the periodic function; thus . 



ff 



(n k } = W~J , 2^7 f (t)f (t+x)dte jnc Vdr , 



23 



or 



$ 



ff 



(n ) = -A=- / / £(t)f (t+r)& 1Tla) k T cifcd.T: 



Also for the finite random function,, 



« ff to> T ■ 






e a dr 



or 



^ff (a3} T = 2T 



-jf' / f T (t)f T (t+T)e~ iwT dtdrr ;; 



_ T «_ T 



then by equating the integrals, since fL(t-) = fi(t) ffxrr 
-T<t<T, it is seen that at a specific input frequency ,00, 



•ff ( VT- 2T *ff ( V " 



where 



ff f k*T ' ' T^° 



2T 



F(i V 



It has been shown that 



*.. (u>) 
T (ju,): -=E- 



£7 (co) 
re 



At the input frequencies, with T large and containing an 
integral number of periods of each input frequency, 

$.(<*>.) $.. (jco, ) m <f>. (n, )2T 

, . . xp k ►. ip k T ip k 

p 3a) k J T. (u>77 ' $. (co, ) m <S>. (n.)2T 
r le k ie k T re k 



or 



p J k 



. *in<V Kn k )P(n k ) F(n v ) 



= i-F k _ 
re 



(n k } I(n k )E(n k ) E(n k? 



Similarly, 



24 



Y f , . . *ic ( V C <V 

Y (DW V ) = 



c VJUJ k' <D. (n. ) P(n, ) 
ip k k' 

This illustrates that the cross-power spectral measure- 
ments need not be made and only the Fourier transforms are 
needed. The latter is usually an easier measurement than the 
former. 

The terms in $ (to) can be examined in a similar manner, 
nn 

In this case,. if measurements are taken at frequencies, co, , 

other than those used in the input, then the expression for 

$ (to) is somewhat simplified; i.e.. 
nn r ' 



nn h 



since $ . . (u),) = . 
n h 



1+Y (jto, )Y (jto, ) 
p J h c J h 



$ (to, ) 
pp h 



It should be emphasized that at frequencies, to, , other 

than those used in the input, 

P(n h ) C(n h ) 

V j V * ETnp- and V^V * P^T ' 

since I (n, ) = T(n, ) = 0. Thus Y (jto, ) and Y (jto, ) must be 
h h p J h c J h 

estimated, as direct calculations can be made only at the 

input frequencies. In order to estimate Y (jto,), simple 

linear interpolation between Y (jto, ) and Y (jto. ,,) can be 
r p J k p J k+1 

used. 



25 



IV. COMPUTER MECHANIZATION 

A. EXPERIMENTAL SET-UP 

The measurement of a human's describing function and 
remnant in the compensatory tracking task of Figure 2 was 
made using a hybrid (analog-digital) computer. The error 
signal, e(t), was viewed as the vertical displacement of a 
horizontal line on an oscilloscope screen. The operator's 
controller consisted of a non-moving force stick. Control 
was effected by fore and aft pressure on the stick; e.g., 
if the line on the oscilloscope moved above the datum, the 
operator applied forward pressure to move the line down, and 
vice-versa. The input, i (t) , and controlled element dynamics, 
Y C (J W )' were mechanized on the computer as were the measure- 
ment algorithms to be described. Each experimental tracking 
run lasted 150 seconds. 

B. FAST FOURIER TRANSFORM 

The pertinent relationships are again 

P(n ) 



and 



p VJ k' E < n k ) 



*nn ( V - | 1+ V j V Y c ( 3V 



pp h 



Now 



i r T / 2 • *. 

P(n ) = -±- p(t)e" :]a) k t dt 

K T " / -T/2 



26 



or 



P(n,) = -=— / p (t)cos(na) t)dt + j/' p (t) siir.(noj t) dt 
K L K-T/2 *" ^'-T/2 K " 



-T/2 r T/2 

p (t) cosCiia)^ t)dt + i/' E 
■T/2 * «*-T/2. 

These integrals may be approximated, by the following, 
summations : 



P(n ) = ~£p(nAt)cos(a) k nA.t)+i ^= £ p (nAt) sin (uyiAt) 



n=0 



n=0 



If 



N 



P t =X)p(nAt) cos-(oa k nAt) 
n=0 



and 



K 



p k =£ P( nAt ) sin.(o) k nAt) „ 
n=Q 



then the fast Fourier transform P (n, ) can. be written as. 

k 



,. v ► At 
( V ^"T 



A + JB 
^ Pk. Pk 



Similarly, it can be shown that 



C(n k , = At 



A + ]B 
C T J c T 
k k 



* «*.# * • At J A "H-jjS 
and E.(n k ) = — e fc _ - e R 



From this, 



and 



Y (JM = 



A. +- f B 

Pk Pk 



"p J k.' A +-]B 

S k. e k 



Y p (J V 



It can also be shown that 



•1 2 B 2 - 

P k + Pk 

2 2 

A + B 

l e. e. jl 

k k. 



1/2 



* Y p Cjaj k ) = taj3L " 



B 



A. 



— tan. 



B 
-31 Pk 



A 



"k 



P-k 



27 



In order to validate the simulated controlled element- 



dynamics, Y (joj) , on-line measurement: of." Y. (joa) can.he^ 



utilized; 



FA. 



!V*V 



z. 



** + 



2 2 
A + E 



L *?k. 



" 1/Z 



Bk; X 



and 



-L p k 
I Y c (jw fc ) = tan. ^— - tan 



-11 C k 



2k; °k: 

The power spectra of the remnant can also: ba r determined 
from the above relationships with the interpolation- process 
described earlier. The determination of the power spectra 
of the operator output , $ (oo, ) , can be accomplished: using 
the measurements p(t) at any desired, frequency [Refi. 6].. . 
Previously it was shown that 

$ Cos- ) i $ &Om = $ (n, )2T ,, 

pp h' pp h T pp h ' 



thus 



$ (co, ) = P(n, )P(n, )2T = 2T 
pp h h h' 



P (n h ) 



2 _(At) 



2T 



A 2 ^B- 2 
P h P-H. 



and finally, 



* t \ * 2(At) 
pp h T 



A 



" . BE ' 
h. h: 



From this it can be seen that in order to determine the 

operator's describing function, Y ( jco. ) ,. the remnant, 

p K. 

^ nn (w h ), and the controlled element dynamics, . Y.'.(jo)..) , the 
only measurements needed are those o£ the: error, operator: 
output, and controlled element output. If each: of : those 
measurements, taken at specific times, is then multiplied 



28 



by the proper trigonometric function, and summed, ov ex the 
entire run , then the describing functions: and remnant can- 
be calculated. 

The major drawback of the fast Eouxiex transform 
technique is the necessity of calling the: trxgon omet r i cr 
functions sin(x) and cos (x) during the run. This? reguixes 
so much computer time that the multiplication and addition 
computations cannot be performed between ana!og^to:-d±gxtaiJ 
interrupts. This in turn means that alL data must., be stoxed 
for later computational purposes. 

The necessity of calling trigonometric functions during 
the run can be avoided if the relationships fox." si_n (a+b) and 
cos (a+b) are utilized in recursive fashion. If the initial 
time of the run is "a" and the time between measurements is 
"b" then 

sin (a+b) = sin (a) cos (b) -h cos (a) sin (b) 
and 

cos (a+b) = cos (a) cos (b) +- sin (a.) sin (b) . 
It is obvious that a recursive process can be mechanized 
which obviates calling trigonometric functions during the 
run. 

The use of this method yields results immediately upon 
completion of the run and conserves computer storage space. 
Within seconds of run completion,, the data may be analyzed 
from either numerical or graphical read-out.. 



29 



T. RESULTS 

The operation of the program was checked by measuring 
the transfer function of known elements or filters^ in. 
place of the operator. The results: are shown in. Figure 3> 
4 , 5 , and 6 . 

In validating the remnant measurement, technique:,, the 
operator was again replaced by an element with known, transfer: 
function. A signal 

n(t} = Asin(a)j^t). ,. w k <a) h <U) k + i " 
was inserted at the output of this element.. Measured 

values of $ (co) were then compared with, the theoretical 
« nn 

| value [Ref. 71. 

Actual describing function and remnant measurements 
were then taken on two subjects.. The first of these had 
considerable previous experience.. The second subject had 
experience only with this experimental set-up.. The measure- 
ment results are shown in Figures 7-18 for controlled 
elements of Y (s) =1.0 and — . Each of these figures 
represents the results of 10 tracking runs.. In all runs, 
the human describing function contains the gain, of: the 
controller. 

The describing functions illustrated are comparable 
with those obtained by other experimenters;: e.g..,, [Ref:.. 8] .. 

This computer program represents a powerful tool" for 
use in experimental investigations involving a human 



30 



controller. It can, for example,, be utilized in a variety of" 
situations in which quantitative models of pilot: behavior- 
are desired. The program itself requires little core; storage. 
This means that considerable storage is available: for." simu^- 
lating complex aircraft dynamics, providing detailed" display; 
formats and calculating performance measures .. 



31 



TABLE 1 



E(jw) 



l(jcj) - N(jw)Y c (jo>) 
I + Y p (jw)Y c (ju;) 



Pljw) 



l(jgj)Y p (jaj) + N(jcu) 
I ♦ Y p (J«)Y c (J«) 



C(jw) = N(jw)Y c (jw) + E(jw)Y p (ju;)Y c (jW) 



Kjcu)E(iu)) 



f ( j uj ) I (j oj ) _ l(jw)N(jw)Y c (jw) 
I + Y p (jcu) Y c (jw) 



Cjoj) p(j cj) 



l-(jcj) I (jcu) Y p (j ou) + T(jw)N(jw) 
~ Yp(juJ) Y c (j w) 



T(jw)C(jw) = [T(jw)N(Jw) + T{jtu)E(]w)Yp(jcu)J Y c (joj) 



E(j">)E(j">) = 



KJcii) - N(jw)Y r (j(u) 



I + Yp(jw) Y c (jw) 



|(jto) - N(juj) Y c (j cu) 



I + Y p (j cu)Y c (joj) 



Pljw)P(iw) 



l(jcu) Y p (jcu) + N(jcu) 



I + Y p (jcu)Y c (j cu) 



Kj^)Y p (jcu) + N(jcu) 
I + Y p (juj) Y c (j cu) 



CCj u») C(j cu) =: [n (j w ) + E(j w)Y p (jw|y c (]w)][n(jw) + £(] cu) Y p (jiu )] [y c (j w )] 



32 



TABLE IE 



Yp(ja>) = 



ie 



Y C (J W ) = 



3>. (w) 

Y iC V ' 



(J). (o>) 



*>> 



= $» I ! . Y p ( j W )Y c (ja/) I 2 - ^,((u)l Y p ( ju# 



pp- 






3>„(«J?IY p (jg,)l 2 + $„>) 

|l + Y p (j*)Y c (jw)l 2 



33 



co 

< 



CD 

c_> 

< 
or 



>- 
QL 
O 
h- 
< 
CO 

z 

UJ 
Q- 

o 

CJ 

o 

c 



tr 
o 

< 

(T 

u 

Q_ 
O 

z 
< 

Z> 
X 





i- 






BL 


3 « 


K 




Z> 


**,- ■^- , * 


O 


o o 












o 






UJ K 






_J 2 






-J UJ 


3 




O 5 




1* 


T u 




cj 






j 


^.^ 






3 S 














Q- 


a 




cr 

Ui 

_i 
_» 
o 

K 

F 

2 

o 
o 






J 


i 
















r* i 


II 
ft: 


o " 


h- II 


i < n 


1 or It 


uj l: 


Q_ 1! 


1 O 1! 
I 1 


j| 
Z 11 


1 < 1 


2 I 


ID 1 


X |! 


1 






1 

1 






_l v 






< < 






=5 -J 






CO °- 






i — en 






> 5 






/ 


V 




cr 




o 


,_^ 


cr 


3 zr 


a: 


■^r ■*-» 


UJ 


Ui * 




( A 


u\ 





UJ 

cc 

Z5 
C£ 

U_ 



3 
CL 



3 ~ 



34 



CO 

< 

o 

z 
*: 

o 

< 
cr 

h- 

>- 
en 
o 

< 

CO 

•z. 

UJ 

a. 

o 
o 



o 

c 



O 
< 

or 
uj 

Q_ 

o 

< 

X 



_J 
UJ 

o 
o 

:e 

or 

< 

UJ 

2 



CO 

< 

O 





*- 






n 


«_» 


CL 


3 -ZZ 


h- 


— 1 ^- 


D 
«*"\ 


u u 




v-^ 








1 a 








3* 




o ^ 


^7 




(T 2 


. o 




t- lu 


>• 




z -J- 






o W- 






U 






j 








**■*- 




3 «— 














CL 


Q. 




cr 






ui 






t' 






.a 






a 






tr 






>- 






z 






o 






u 






y 


t 




-— (+) ! 


<*3 V y 


zi,r / 








5z "■= 


3-S 


o: 


!?"* •- 






a. 


a. 










1 


ji 










""a. 






> 

1 
1 


■ 




+ 

1 
1 

1 
1 






i > 






ISUAI 
SPLA 






> 5 






; 


^ 




«r 




o 


»-» 


cr 


3 — 


cr 


^^ — -* 


bid 


UJ « 




M 


^\ i 





C\J\ 

UJ 

tr 

COS 

UL- 



a. 

Z 



3 — 



35 



20 



S o 



-20 



-40 



-60 



^. 


45 


M 




<D 




Oi 




w. 




o» 





0> 




•o 




*•* 






-45 


3 








. Q. 


5 


-90 



MEASUREMENT of KNOWN (-L-) ELEMENT 




— Theoretical 
° Measured 



-O O — G Q £L 



O 



-135 
-180 
-225 



J I r i r i 



J L 



.1 



I. 
CjD (rad/sec) 

FIGURE 3 



36 



MEASUREMENT of KNOWN (- 



S + 



-) ELEMENT 



20 



€ 



—-20 



-40- 



-60- 



J-_ 


45 


U> 




0> 




Q> 




v. 





<u 




■o 








^^ 


-45 


3 





-135 

-180 
-225 









— 


— Theoretical 
o Measured 


1 


1 1 1 1 I 1 1 1 


^ — . ___ _o_ 


o o G 


r 1 1 1 1 1 ! 1 


I II! 



.1 



'(a) (rad/sec) 
FIGURE 4 



10 



37 



20 



~ 

■o 



--20K 



-40- 



-60 



45- 



m 
' -45 



i' 90 



-135 

-180 
-225 



MEASUREMENT of KNOWN ( — «-) ELEMENT 

v (S + l) 2 ' 




— Theoretical 
o Measured 




I I l I l i i 



J I i i i i 



I 



w (rod /sec) 
FIGURE 5 



10 



MEASUREMENT of KNOWN (_— ?— _) ELEMENT 



20 



3 



CL 

--20 



-40- 



-60- 



.^. 


45 


</> 




Q> 




Q> 




i_ 




o> 





Q> 




•o 








.^ 


-45 


3 




^«— ■* 




D. 




>- 


-90 


vj 





-135- 



-180 
-225 



.1 



S^ + 2S + 2 



n n r* ■** 








— Theoretical 
o Measured 


/ 


i i i i i i i i 


1 t 1 i 1 i l 1 


o 


i iii 



oa (rod /sea 
FIGURE 6 



10 



39 



II 
In 



Z 

o 

H 
O 



e> 

z 

m 
or 

Q 



< 

X 



UJ 
Q 

-J 
CL 



O 

UJ 

GO 
CO 



f-QW 
K* 



KH 
KM 



KH 



KH 
KH 



H>-f 



KH 



Ka- 



KH 



KH 



h<H 



CJ 

in 

3: n~ 



o 



(QP) !(">) Al 



O 

CM 

I 



40 



o 



U 

3 
u_ 

O 

z 

CD 

or 
o 

CO 
LiJ 

Q 

< 

X 



UJ 
CO 

< 

X 
CL 



———————— 


; 
i 


KB 


D 


— 




















— 


Ui 
CQ 

en 


Q 

o 

Q 






— 




Q 






- 




O 






- 




O 






— 




f-OH 






























MD-} 








1 


i i 


1 


i 


4 



-5 °° 

0) 

•o or 

-. Li- 



en 



o 

(S98j63p) (m) X "V 1 " 



o 

i 



o 
■? 



41 



trt 



z 




o 




t- 




o 




z 




3 




u. 


— 


z 


1- 


CD 


UJ 




~> 


tr 


en 


o 


3 


to 


CO 


UJ 




o 





3 
X 



Z 
< 

z 
:e 

UJ 

or 



-€&- 



H — G H 



-Q- 



H o 



-o 



f Q 1 



-O 



-o 



-o- 



a 



** a> 



o 




a> 






UJ 


■D 


tr 


CJ 


n 


i_. 


e> 



o 




o 


o 


o 


CM 




Tuu 


to 


CO 


1 


(qp) 


(*>) $ 


1 


1 



42 



V) 

>> 

v. 

o 



Z 
O 

o 

3 



o 



CO 

a: 
u 

LJ 

a 



3 
X 



UJ 

o 

3 
H 

_J 
Q_ 



CVJ 



O 

UJ 

—> 
CD 



HDH 



hGM 



l-O-H 
HGM 



K>H 



KH 



KM 



D 



a 

<u 
a ID 

* LU. 



h-OH 



i a 



o 



(qp) 



lH d AI 



o 



o 

CVJ 



43 



II 



o 



Z 

o 

I- 
o 
z 

u_ 

e> 

z 

m 

cr 
o 

CO 
Ixl 

o 



3 
X 



UJ 
CO 

< 

I 





fa 
P 


CM 

H- 
O 
UJ 

~5 

CD 

z> 

CO 


K>-» 

a 

D 


1 


l—OH 

i I 1 1 



u 

to 



a 



UJ 

a: 

3 
— C5 

3 u. 



o 

CD 



o 

0) 



o 

d £ 
(S99J63P) (p>) x~r 



o 


O 


h- 


ID 


CM 


ro 



44 



o— 



V) 



o 



I O— i 



=Cfc 



-O 1 



z 
o 

h- 
o 

z 

3 

u_ 

o 
z 

GO 

a: 
o 
en 

UJ 

a 



CO 

\- 
o 

UJ 

-5 
CD 
3 
CO 



■o 



O 



o 



o 



(M 

o 
a> 
f> LJ 



o 



3 



< 

3 
X 



H 
Z 
< 

Z 

a: 



:Q: 



O 



O 



-O 



-O 



o 
cvj 



o 

CVJ 



(qp) 



o 


o 


O 


UU^ ^J" 


CD 


CO 


(en) $ • 


i 


1 



45 



-I 



CO 



II 



U 

Z 
ZD 
U. 

O 

z 

CD 

tr 
o 
en 

UJ 

Q 



I 



UJ 
O 

_J 
Q_ 





f— 


D-H 


- 






hCH 




- 






KH 






o 




O 




- 






JCH 




- 






K* 




_ 




1- 


hCH 






o 

fcO 
T3 


o 








O 


UJ 


hOH 






v_ 


CD 










3 
CO 


KH 






3 




l-OH 










1 O— 4 

i <^ I 










1 o I 

1 <"k 1 










1 <^1 


i 








1 o 


i 






1 


! 


1 


1 





ro 



UJ 

cr 



o 

CVJ 



MP) ((">) Al 



o 

CVJ 

I 



46 



fOi 



m 



-I 



ii 

V) 



>-° 



ffl 



o 

»- 
o 
z 
I> 
u. 



o 



m 
o 

a 



X 



in 

< 

x 



13 



101 



d 



U 

UJ 

-> 
CO 

CO 



f3 

m 



la 



1-04 



O 



o 



o 



o 




a; 




<rt 




\ 


Ld 


-D 


cr 


O 


Z> 




o 


3- 


b- 



o 



o 

en 



I 



o 
a? 



(seaj6apj 



o 
d JS 



o 



o 

CO 

•? 



4 7 



-I- 



CO 



H O 



H 

z 

Z> 
u. 

o 
z 

CD 

tr 
o 
(/> 

Id 

o 

z 
< 

X 



o 

-> 
CD 

CO 



< 

z 

LJ 



-o- 



I — O — h 



£t 



cr 



-o- 



-ce — H 



H-o 



o 

w 

o 



in. 



UJ 
O 



I Q- 

l o 



-O 



-O 



-o- 



-o- 



o 



o 

CM 



(qp) (^) uu cj, 



o 

CM 



O 



48 



h-O—f 



ho-\ 



■f 

II 

7? 



hCH 



KW 



O 

O 

2 
3 



O 

2 

CD 

E 
o 

CO 

UJ 
Q 

z 
< 

X 



*3 
O 



CM 



o 

UJ 
CO 

C/J 



fOf 
KM 

hCH 
h-O-f 



(£) 



3 W 



UJ 
Q 

H 

_J 
Q_ 

< 



-o 



l o 



o 

CM 



4 



J 

o o 

(qp) lC>) d AI 



g 
i 



o 

CM 

l 



49 



h-O. 



m 



+ 



en 



o 

I- 
o 

z 



o 

z 

DO 

q: 
o 

CO 
UJ 
Q 



3 
X 



CO 
< 

X 
Ol 



P 



a 



p 



a 



CM 



o 

UJ 

m 

D 
CO 



©i 



N. 



o 




a> 




to 




v. 


UJ 


•o 


DC 


o 


Z> 


l_ 


o 



KH 

1 



-Q 



O 



O 



o 



o 

(saaj68p)' 



o 

CD 



o 

CM 

i 



«? 



50 



\> a- 



¥ Q- 



1 O 



-r 
ii 

5* 



o 



« — o 



o 

2 

o 

2 
CD 

a: 
o 

CO 
UJ 

o 

2 
< 

X 



CO 



<\J 



a 

UJ 

CO 
D 
(0 



-O- 



■O 



a. 


— 


v 




in 




Xr 




TO 


UJ 


a 


cc 


•w 


"D 




o 



-o 



f Or 



-O 



2 
< 

2 

UJ 

cr 



-O 



-o- 



-o 






o ° 



o 

CM 



O 

i 



51 



.-i — — • >0 £L Ul O" 7) C3 

^ V IV If) «s X U! Ul b >- K Ul 

x "«— — «-» x _r ar: co or cj cr 01 

* «-t O U. -* ^C L7 < Li_^LuLi_ 

>cn>-«-«*-* LUX Ul > 

in x x «. r^ x •» o o_ co :t> co 

• if; lO * • CTi X X Z* G» UJ 3 — 

O ^ « CIO •«. «% 7 ^ CO Ul CO CO" >- 

«-«xxa>«-«'~-or«-»cwi 3^ 01 < ry ul 

b_ O O uJ U. — > Ul ■» U1 U. X lU Z 

«-t «h «-• z «-• — ~-c co x :> a. >• ui- 

> ^ «, to * 3- c — m ur 75 

X ~ -— X u JZ £L C*l ZZDZTUiCJ 

%0 IV M vO >- •— <■ ■» Cj CO CD 'CO LU 

% «■ - — «s CO X U_ »— X. i»+ (K »**■ *C EC 

.^•^•O-XUJCLCn^ •* t— LU H- X U_ 

— • < <: j- ^ x a cd u>cjo_ 

n X X «* -i: GO Lu O 7!! Z" CO" 

x in cl r^ x -v a? uj 3 to. 3. or. d 5*- -*■ 

r^ % x ♦ vo x r> co 7 u cj l_ cj co co c> 

— xr«*.ifi%oocn<:*-» ►— ►— or. en oj — 

\ O * — « X - <. I U I (^ X." kl Jc — Lfi 

v. «-t x u. ■* — lj a_ co z < z: «c >» i— -— cxj «-• 

•» «. «-» i«c «— i — ? jr j. »— ■— » J 1 •— * z «• c" uo *■"■ 

— ~ — .» .* rx -~ j- a> >- to. >*- ae x oj •»- lj 
y VN" i • o. <cu a •— a — • - o lj co — ■— a. 

— —»^rr'X)CL ■* cl or or 2 ljop x 
3 Ll.lt — :<*-—» — to X •— (J o 1 J C\ a- — 7 «*>Oj ■. 

X CO »-• Q_ Ct Ll X •—CO X CO Ul CO UJ CLO. CL7 — — - 

* iil>- a--« a ma m _t u j •»• ««• c . oj 0^ in 

*» LT1 >£> X Ul •» X "S «• O _J O _1 t — t— X OJ <£ OJ -cH 

X -s-.lt. X ■ -X. £_ X •• CD Z •»- CO » — — 

oj xx-. ox ■* _j ■* cj t— cc *- rr <i- oc L3 — 00 

w o oj x «» ^0 x lj — x ea y— cb i — zr_i r «*»o> < x 

■v t-» ^ o x cc r* oj f— _j z _j z r: ui lu — ai x .n 

— ^ LT! «-i -* Cr: % _; ^ •— cr- •— -rr UJ c ■»• . — o_ •» 

V ^^-^ ^. jjj — OC D_ C_ CJ CL— LJ DC «r. C5 03 -~* h-^ 

**v— — >~-»>.x~>uj:>>- Q <r+- CO ~-CD LG 

0_ — 1 V ~» * ^C — ^— — Lu U. SjLl ~Z U_ U_ C/3 f^ rf «« CTi *h 

x lj \ - n .. ■. :-• co ** o en c oa c en ? < •• — — ro — 

<c < : \ 7*. • «-• x x •-• ¥- o lj co ►— -~- cr. en — -o 

"X X ^ T i(" U C N . ^r _J _J U 7" X X UJ X X U c: ««■ C" OJ C<1 U > 

1^ >C ^1 - 4 ^ H fL ^ Sj 0_ — Q- - ^ r . 0. — ~0L % •■ c ■ OJ — ^- > < 

• «. s V -. U * ^ X ■< > < - l. <t <: X -tt. .' ■ <s »-> 1 — lo ^- «— ^ LJ ■» •> 

IT; X X — X iO II X T-. I K- r^ 7 TtL fV! ;,") f>' 7 T O v |.U < < k - '- 

«-» O C CL th •» C 7 ■» »-< — < <. C: C: C3 <7 Z :"' L CT" s U: ^: > ■» ^ C N LO 

U *-• «-* C2 ■«-» If) CO I s * ^ — O C O X LJ "1 X <f X LJ X «-> 7 rr: ^- ^- -^ OJ — • 

*-t «, *. x- •» «-» i— ■ — > r^ x k- X ~ c u: OJ LfJ — r U! o> U : -^ r en r~ — — 

■s n: V J") V — 75 -O X • 7 ^ CO \C CO vU — r^ O O CO ro * •» 3 OJ OJ OJ 0- X 

If) I X ^ X ■* Cl <h riior n 0". Ul ■* X ■» c * < * x •**—■ UiCO <r> 

•-^ «-tX«-i\XU -s^-.«» Lxi » '_ X CO X ." XCf XCIXm ■> CfT "■« Ul Cl X C: 1 

o -. ■«. o ^ x. va *-» "* >: u. x ^j x r* o 2" oi cr rv cj ^. c j 3 cc : - ■« 1 .i_ fi •» 

»-» x x x-i x •>. -. » r^ -^ 4- o •— c> ~ — n -r n lj (u :; r 1 -i ro v •— u^ — % -> -v ^ 

U. 30 -* •» ^r -— >; u • •» •» C • _i (*ia.(\j «•-* »Il »•- **- **3 h(T. «^«u 

•r-t •> - — •% v -r C* ^ ^1 X L.' ■» «- - I . •— X ._i X J") X X /.: X i— LL C3 CL L7> C> 0> -^ 

« ^-. «-«Sf>v'S-*C0«-'X*-« — 3 — « x -r \ \ x. >- \ -r \ . *■ CO — Ci TO O 1 — 

xt x — \ l; \ cr u x ~ 7 • t. J" < \ '.r \ co \ o \ :r \ "7 *— t— a. — -— — n 

c . ^ ^ r.' \ < . \ • «-. \ - — - ^ ^ , r \ v. ,_ X X X X r: X Z> : L U! £ >- 

— — -^ 'i — . — 1— - — X — . — — — ^ — <— CJ^-li.'- ll a < •- U X < 

— r_ 11 \ _ — • — •— 

h->~'-»— LO»— Xt— v0>~*— X»— \C ►—•-•►->— H- .O t— f K h- Z 9, V Z "J*" Z 7 ~/' y 

< < rx <a - < rx <. ^ (o < •> < ro -r < < > <.t < -: lj < >~ ■< a .7 e. 

^ ^ • r. < : -» . _ x ■-.'-- — r ■» _ < x ? o T. or: : r : lj : _ " : r m 

a ;r ur! : l. r- x r. * ■< cr ~) ry x >- ■ _ . u. cr. . ' ." >" ." 5 >: " 

T CO T-» ; «-i ~- ».• C »i h- w — C • r*" "* - ■< '.'. ■ OU f Z O . t. " . .• . C . " £ . " 

U U U U -. U «. Ll r^ 7 U L, L. '• IU U L J u. »— u. t U. -- U U L. U LJ LJ' LJ U L-* 

«H •r-i «— < •-< iV »-* »-• iL w-f »— > «— ' «— ' •— t 

O ■r-' Oj P? ^f L0 vT 1 fs r» c " >— * 0J ( . ^J" 

O OOOO OO O O vt «-» «W «H >- 



52 











in 
























































in 
























































in 
























































o 

• 
























































^ 
























































in 
























































o 
























































o 
























































a 




























en 




























x 


















en 










en 




























-3- 


















z - 








»■ 


en 




























^ -^. *m. 


















K-t 








_l 


•* 




























moo 


















J: 








_J 


o 




























in o o 


















<: 








■< 






























id o in 


















u 








»— f 


ti 




CO 
























OrHN 


















to 








t — 






>- 
























• • • 


























■ — 1 


Cf 




<r 
























^ ^. «« 


















t— 








—7 


CD 




cc 
























n m <t 


















Z 








*-* 






rr 
























onro 


















< 










+ 




< 
























o o o 


















Z 








Ui 






























a_ a_ cl 


















2T 




o 




t— 


Z" 




33 
























~7" ~r ~t~ 


















UI 




•• 




< 


< 




o: 
























-4- -4- -4- 


















Of 




o 




h- 


X 




UI 
























^ •» ^ 


























in 


f— 




i^. 
























in o o 


















cc 




UJ 


































moo 


















o 




~5 




>- 


rr 




a 
























■s> o o 


















U_ 




_L 




o 


Ui 




z 
























O CJ lO 






















< 




< 


f— 




< 
























• • • 


















K- 




>— > 




— UI 


< 
















U! 












^ ^ "V 


















—y 




r> 




en »— 


UI 




Ui 












Q 












--H «-» OJ 


















a. 




CL LlI 


*-,- 


OJ CO 


ne 




>'_ 












3 












o r? o 


















-t.' 




7 "v 


C 


— 


C5 




•— 












t- 








-~ 




o o o 


















•— •« 




M < 


(XI 


*-• 7 






t— 




















c^ 




LC L 






















X 


•fr 


ll •—• 


u_ 




•V- 












_J 








^ 




XXI 


















^ 




La 


—+ 


M 


3 




i — 












CL 








m 




-?• ■* -? 


















-t 




m •• 


It 


•^ :r> 


_1 




~" j 












?". 




— 




*-) 




•> ^ % 


















X 




t— CTJ 


V 


+~~ fc— 4 


<f 




CL 












<: 




in 




3 




IT C C 

moo 

ir. t_ >c 






















J-7 


•* 


— > 


> 




^ 












»- 




y 




^ 




O 0J •>-< 


















IU 




<T Ll. 


*^ 


■» _1 


7 




rr 






'— 






< 




2 




(\: 




• • • 

"» K ^ 

<~> O 1^ 


















CD 




uj in 


*■ 


V <D 


*1- 




en 
u. 






cr 

1— 






z: 

u 




d 


•— « 


>- 




o m «-i 






















UJ 


~_f 


CL. cL 


A- 




to 






a. 






DC 




^ 




X 




O C O 














^^ 




>- 
U 


- 


ir >— 


^"~ 


• 


h- 




u 












i . ; 




in 


}~ 


^ 




^ ~ 3 












o 


O 




'7~ 


[j 


•« *rf 


>^ 


^ < 


'(D 




__1 






•> 






2f 




«-< 


<i. 


»-« 




^- -3- ■* 












■* 


o 




Ll 


f'. 


03 _i 


< 


XL X 


• CO 




«i 






h- 




r* 


i — 




»— 


^ 


:o 




— — — 










1— 


■3- 


o 




— j 




r 7 


— 


-— 1— 


c 


in 


> 






~j 




ai 






a 


5 


^ 


U! 




m 


CV 


.-* 




—> 


i 


•c— 4 




C3 




lU ll- 




*— 


CC I — 


ru 








CL 




0^ 


CO 


CJ 


ta 


5 


-:<- 


1 


1 — t — #— 


+ 


m ip 


+ 




~1 


• 


— - 


■ 


U 


*-» 




— « 


^ r 


O Z3 


,-< 


_J 






^ 




in 


u ; 


■«-• 


•— • 


Ui 


«— • 


-■: 


CE X C ' 


w 


O 3" 


IS 




x: 


* 


I— 


rj 


»v 


■ 


ii- 


o 


O '-0 


«-• Q. 


• 


<f 




o 


»-« 




«-H 


_J 


• 


^* 


Of 


— 


-» 


fl _ 0. 








O 






u' 


* 


u 


o 


'^t. ; 


Q 


■.-■ 


i y. 


c ■ 


•— 


c 


• ' 


«-■ 




-7" 


<f 


c 


o. 


. . 


~ .• 


V- - 


I — t — •— 








♦— 






- *-\ 


CJ 




»— * 


■S" ' "• 


iH 


•< 1— 


* — i 




H 


• 


v^y 


u 


ft 


«— f 


I I 












. 








. :' 






jj 




J 


-• 


*-• : 5 


• 


<.c i__: 


it 


n 


> — i 


(. ■■ 




-c 


c v 


< 




II 


7 


Z" 


7 


^-> 


co en to 








*— • 






X 


ii 


•— < 


IT 


«~ 


•n 


— en 


CO 




_v 




t! 


f5 


in 


rn 






c 


*7 ' 


c 
























— 


t— »— 


— • 


Ui 


t— UI 


»— . 


•— • 


ii 






^ H 




•— 


•— 


5 


i: 


11 


_1 


i ! l f 


.< 


<: "3 


j* 


1- _( 


^< 


X 


_l 


*— 


;--* 


O 


'.". _j 


G 


\— Ui 


U _J 


to 






ft 


_J 




11 


_i 


_J 


> 


2 


X 


_l 


1 _J _» _l c 


h- : 


jj» 


_J _' 


ci 


L> 


_l 


_) 


< 


< 


, 


<r 


— '•" 


i/: < 


• 


f— 


l— 


t— 


l 


II 




f. 


c 


». ' 


c 


~i ' 


<-. 


<"<.<. 


_j 






_T -i. 


_i 


M 


< 


,_! 


Ui 


UJ 


»-* 7 


Lu 


a — 


^: u 


~~ 


UJ 


— | 


D 


■t; 






" 


^ 


o 


o 


CJ 


u 


u o l; 








U 






u 


o 


-v 


. 


Lu < 


[ v 


j- »— 


<r ir, 


I 


to 


D 


Q_ 


u 


h- 


r\ 


<f 


. ; 



co to to 



f to 



53 



u 


































<D 


































co 


































Ui 


































Q 


































Z> 


































»- 


































►— f 


































^ 


































C3 


































<c 


































i: 


































Ll! 


































Z 


































»— « 


































51 


































fr 


































Ul 


































k— 


































Us 


































Q 


































or 


































a." 


































»— 




























. 






< 


































_1 CO 


































Ul 
























CO 










s -~ 
























<. 










<: u 














U 










LU 










u ;' 














< 










en 










ui 














o 










-«. 










<n z> 






«~ 




*— 


»■» 








""' 


^ 


2 










to O 






a 1 


--- 


CVJ 


V 


<r> 






Li 


d 


CJ 'U 










Ll 




--» 


* 


»- 


* 


-^ 


f— 




— 






/ r^ 










• cr 




i— 


x 


Llj 


* 


CJ 






~^ 


o-: 


or 


CE 'X 










CO U 




_1 


■^. 


CO 


^ 


* 


Ld 




^r 


o 


fT" 


in k 










O 




LU 


V 


2 


y 


*■» 


Z) 




U. 


u_ 


U 


L- (E 










U _i 




a 


— 


CJ 


"^ 


V 


_J 




Di 






i— 










cr _J 




9 


< 


* 


V ' 


— 


<. 




'."~ 


►— 


»— 


>— CO 










U «i 




^~ 


i 


•»-» 


i 


D. 


>• 


i-^ 


^ 


) 


j 


./ 














V 


• 


V 


• 


\ ' 




CO 


CJ 


d 


CL. 


Q 1 










_J H 




~-^ 


«-! 


»— 


•«— < 


<£ 


_1 


* 


* 


t— 


h- 


I— _1 










_J < 


0> 


"v 





j? 


*- 




< 


2D 


•— < 


~» 


Z> 


3 <- 


0> 








■c^ 


(u 


^- 


y- 


»— 


h- 


+ 


■— < 


CJ 


■ 


j"~ 


jj 


C 


c* 








•J) 


■» 


. 


(X 


>r 


cr 




»— 


* 


D 






Ul 


■^ 


• 


• 


• 


Ui h- 


«H 


»■— < 


* 


•— « 


■ 7 


r— 


>— 


*— 


% 


CO 


CO 


CO rsj 


>— 1 


: 


' — . 


o 


^ii . 


II 


-0 


CO 


CO 


v~ 


TO 


LU v- 


— \ 


<£ 


LJ 


1 . i 


LU •— 


II 








— . Q 












u 


' •— * 


u*_ 




i — 


r — 


t— _i 


^ 


ll 


II 


II 


5 >- 




II 


H 


ii 


11 




~s* 




11 


c~ 


T) 


tt ■*( 










cr -> o 










ii 


— CO 


•1 




i? 


/. 


i " . — 


c: 


*-~ 


-~ 


■~ 
















t— Ul 




. w 


LU 


ud 


Ul ►- 


Cvj 


V 


V 


v: 


k 




V 


•y 


r* 


V 


f- 


. 


v- 


>— 


C^ 


o 


c: — 




-~- 


-^- 


— 


U' 


C! 


^- 


*- 


«~ 


«-» 


70 


C ' •— i 


^ 


Z> 






—r 


-*. 


U) 


U-. 


a. 


Q 


O. 


< 


CJ 


CJ 


O 


G_ 


U CD 


c 


L_ 


Uj 


«-• 


CVJ — 


L7: 


^- 


•< 


< 



>^» 

o 
<e 

LU. 

an 

(X 

<n — 
i — ^ 

<Z "O 

or tr 
ui 

IUC0 

i— 
a 

lL X 

f- CJ 

K I — 

<*: •-#- 
co (n 

Ll! O 

m co 

o 

o: 



LP 






>: co 

lu — 



on o 






— co to »- 

«-» Ul • ~ > 



O O O O O C) C^ G • O 

lj r^. «~* 

ii ii ii _j it ii ii it ii — 

~r » ui »— 

^» ^, ^ _ h- o c? a n> _j ~o 

v v _v t— z i— co ld co to . 3 a 

— — — , 3 »-*»- cr U! c . «4 h- 

aJ »-* cu c v'- _> .3 rr i; •• x _o 

1L Cu. vj a. i L-U,C KLo>: 



CO 

_J 

u 



o 



CO 

i 

U 

«— » 

CO 






CJ 



54 



<r> 



















— » 






































to 






















tr> 






































CO 






















CVJ 






































m 






















^ 






































_i 






















«-i 




























































n 






































— cu 






















2* 






































« "■■ CD 






















^ 






































— — u_ 






















^~ 






































— V 






















V 






































N^ ^,f_^ 






















-^ 






































«— CU CJ 






















CM 






































•H < !_d 






















£11 






































■u ^- rr 






















^ 






































■k— rr 






















•^ 






































~* v: (r^ 






















¥ 






































V — CJ 






















^ 






































— CVi 


Ui 




















cu 






































t-» a cd 


2 
















^ 




<c 






































fJD — ►— 


* — • 
















o 




> 










UJ 




























— z 


1 — 
















to 




^-» 










TO 




























Z <f CO 


~> 
















o 




V 










_J 




























<" »— ui en 
















2 




— 










<r 




























V- <£ LU 


DC 
















r— 




T-l 










> 




























< K 


















^ 




cu 






































l: (.0 


h- 
















rv 




•>. 










U 




























Ii LU 


Z 
















■s 




— 










5C 




























— Q 


UJ 
















O 




NT 










< 








, 




















^» ^- 


CO 
















CO 




— 










3 




























— V o 


^_ 
















LJ 




«-• 










O 




























V — 00 


<: 
















z 




< 










to 




























■^*-- v— i t-h 


t— 


cu 














<r> 




«i 






































LU <f 


i 


* 














^ 




^ 










7 




















a. cu cv 




<t «r Z 


c. 
















vC 




M 










<c 




















* 


* 


* 




^ — < 


:T 


t; 














* 




«^ 










LJ 




















* 


* 


*■ 


«-« 


• V X 


•< 


»— 














C3 




LJ 










> 
































• 




«-< 










C^ 




cc 






























^ 


V; 


Nl' 


V 


V — T-. 


-^i 


S3 O 




ro 














> 










(0 












z 


y 


z 




UI 


•H 


a. 


UJ 


n. a ~- a; 


•— • 


(0 sO 

0" 




S3 










LU 




V 










23 




Z 1 


— ^ 


^ 


7 


UJ Q 


CJ 




ro 


cc: 


ca 


u 


U. «" Z lU 


^ 


*— 




K- 










«* 




^ 




r* 


^~> 


r • 


Q£ 




o 


i 


o 


















\ 


\ r < j c: 


»• 1 














JO 




U! 




co co 


Lo 






CJ 


U! 


u 


U 


li. 


''V 


rr 




+ 


+ 


4- 




— -j. i — 


— i 


o 




<D 










•* 




<: 




H- 


f- 


i— 


LU 






CO 


CO 


CO 


; <.. : 


;, 


:;■; 










V 


V t- <l o 


i— 


c>. •— 




CO 










c. 


~%. 


•s 


• 


13 


; 


3 


I 




3 


2 


^ 


"" 


u. 


Ll 


U. 




re 


r<i 


r^. ; 


»— 


— - <c — < 


-:] 


— ^ 














CO 


O 


V 


-? 


C. 


c 


LC 


1— 




Vt 


i . 


a 


cc 










+ 


* 


* 


r~* 


Lj — * Uj 


5" 


• - — 




<-» 










f- 


o 


— 


* 


\ 


\ 


\ 




• 


* 


* 


* 


* 


2f 


Z 


z 




* 


* 


+ 


u 


a- * ro _l 


r 


t- cv 




«-( 










7_J 


o 




o 


f • 


f • 




CO 


o 


c 


r - 


( 


r * 


fi 


c 


^J 




-^ 


— ^ 


*-~ 


»^ 


— ro .. 


c 


_J «< 




• 






lU 




Q. 


« — i 


• — 


CO 


L0 


l0 


CO 


UI o 


CO 


CO 


CO 


C/) 


<— • 


.-, 


►— 


s . 


^ 


y 


^^ 


1 — 


■— • r*" ui 


u. 


• _c 


U' 


r *- 






_J 




^. 


•»-» 


re 


- 




u 


•• 


! 


c ■ 


I— 


'v- 


u 


c 


h- 


i— 


h- 


(\i 


— 


•»^ 


— 


, V 


r» r*» j to 


7' 


<— □ 


_J 


Li 






ro 




^ 


»— 


o 


a 


cc 


z 


■ ; 


<: 


«— < 


— J 


or. 




:x 


<■ 


<; 


<. 


>- 


UJ 


^-t 


ri; 


' ' 


C3 00 <- 


•— t 


V 


C 


• 






<c 


_ 


»-- 


.- 


*-i 


UJ 


u 


c 


t- 


^J 




L 


w*. 


(T 


t— 


k— 


i — 


i — 


»-l 


<t 


^ 


< 


CO 


CC H » I 




~- II 


< LJ 


1 — 


o 


^J 


i 


_J 


:<: 


CO 


V 










CO 


u 










*^^ 


■f^ 


O 


II 










u 


UJ 


' . 


~S -v 


•— . 


; r 


__■ 


i — 


■. 


i j. 


-;. 


N«_ 


ii 


II 


•i 


11 






ii 


M 


ii 


ii 


^_ 


2 


< - 


^ 


ii 


II 


ii 


II 


II — - — » 


CO 


<i ■— 


u: rz 


TO 




7 


^-\ 


~z_ 


< 


cc 


^— 










7» 


T* 


























V V LJ 


*c 


07 V 


*— * 


^ 


c 


— • 










Lu 


C3 


>- 


> 


V 


23 


CT 


Q 


:i: 


C3 


t_' 


Cj 


CO 


CO 


(.3 


--> 


^ 


— 


*^ 


^^ >^ — y 


. Ii 


a. — 


_l H- 




1- 


t— 


_J 


_I 


_! 


_J 


t— 


CO 


CO 


CO 


CO 


CJ 


U 




;~ 


.0 


CI 


— > 


«_1 


•— ► 


^r 


?ii 


V 


^ y 


V 


\' Q_ LJ D 


cc 


— cv 


_J .' 






1 ' 


i 


_J 


i 


_J 


* — * 


ry 


. 




c 


C 


CO 


I— 


.r 


LlJ 


2 










»— 


■ — 


~- 


«^ 


^- <x < CO 




< 


<: o 


u 


c . 


c 


< 


*c 


< 


< 


^_*' 


Q 


Q 


z 


_ 


%■ 


» 


73 


r 


I 


5 


-i-l 


r- 


c_< 


3Q 


UJ 


CL 


CJ 


Q. 


U X X CO 




U. X 


o u 


•— • 


C 


O 


u 


LJ 


U. 


c,' 


-* 


Li 


Lu 


^-_ 


»- 


q: 


a 


a 


UI 


(D 


t- 


U 


l-L 


i_^ 


Ci 


u. 


u. 


c^ 


>- 


>- Ci_ U_ -i 




— u 


o 






•r-H 






















































m 






m 























































* * * 



** *■• 



55 






-3" 

CD 



m 

■3- 



O 



<n O 
CO nO 

oo 

• i 

o 

00 *-« 

• - — 

t- o 
_1 < 

• X 

^^ f 

V " 

— tl 

u 

< — - 

T V 

x — 

— CJ 

< 

X X 

•— X 

(XI 



(0 V 

~ X 

• *C 

O I 

,: X; x 

I + 



to o 

• sO 

V 

^- II 



T V 
X — 

< 
U_ X 

*~* Q. 



o — 
u 

«-» <• 

• X 

c x 
I 

• * 

»- o 

o -o 
• r> 

V it 

— LJ 



O 

>- 

r> 
< 

a 

>- 

u. 
cr> 

en 
z 
jj 



x 

0. 

>- 

< 



< 



0> 

ro o 

c> z 

i^ < 
in 

r^ _i 

* UJ 

O Cr 

O U 



m x ro m 
• • • • 
r^ r- r^ r^ 
in in in m 
\ \ \ \ 

V V V y 



x 
<r 

X 
X 

C0 

LJ 



u u 

< <r 

X X 

a x 

CO 2 

C* •— i 

U uJ 



V \ V V 



n 

>- 



u ~ 
<r v 

x u 

^ <r 
u X 

—• x 



. 



U LJ 
>- >- 



Z 

a 

CO 



7? 
< 

UJ 

a: 



CT! 

7 
CO 



3Z- 
X 
X 
UJ 



or 

CD 



II tl UJ 



— — ■ r j a io y v 

t- I- lTjJ * 

z z z t- o_ o 

3D CD <Z. UJ CJ > >~ 

U (J L ij t' < CU 

n c: 



y y 

lj lj 
>- >- 



U) 
CO 

_> 

rr 

(C 

C.J *-i 
< «, 

U. x-l 

LO 
CO II *-< 

I_> V -*• 

Z Ul 

— • t- si) V 

f— <: ■* 

z _i . it 

c. <r c 

OX £_. ~J 



•H^« — + 

+ <-> U M 

^ -^ >- — 

~ + < C_) 

o y >- 

>- ^ + CQ 

< C_> 

> -- + 

— + u y 

— ^ < o 
u y ~ > 

>_ ^- ^- CD 

< u — 
-^ >- * <— 

— .'Q 

^- ^~. * 
* — UJ 

t— — < 

^ * <: ui 

Ul _ll- 
t— --» 13 < 

<tuQ J 

_J »— * CD 

cr> < ^ a_ 

~- I — * 

^^ * c % -^-< -^ 
y "- X + ^ 

— — * y; «-i 

-•i ^H — — + 

+ ~ X V 

I V --* >- — 

— + 3G. 
^ X _V > 

»-!>-— + < 

+ < X 

V > — + 

— + X) V 

-- + a. y 

\ V > 

— — — m a_ 

— x v — >- 

V V — ^ <c 

:.< — > ~ — 



i ^ ^ 



— + 






— • I CO — 

:- --< * u; 

— -- ^ * f— 

^~ ^~. < 

II II uJ — _J 

H- "^ C_> 
Ixl — » < UJ X 
l-y JH * 
< «^ c ■ < '- 

_> (. ; a _' — 

x 5 a + 

*-■ cu r. 



cu 

CO 

z 

CT3 

c_> 



— Lj 

♦- X 

* >- 

• * 
OJ — 

— V 

\ — 

■ — * X 

~3 a 

X X 



^ 00 






^ ^r-l 






«-l -\ >- 






-. «-< CO 






H II (D 






M -> ^ 






V ^ V- 






^. ^* ^ 






^ ~> 5- 






V — CO 






-UUJ 






LJ X Z 






<C >~ OD 






X -N •» 






X ^ >- 






«» ~> CO 






— — X 






v ^ rr 






— z Ul 






LJ — ^. 






>- X C3 






"* X v/) 






-^ -* CD 






y — :< 






— ~y »- 






x — ■» 






< X C3 






X X CO 






X — • IjJ 






t X z 






— x m 




^^ 


V «« •« 




•-» 


■« 1 ^-^ '""^ 




y 


a X) cr. 




— 


>- T"« X 




X 


•* + X 


*-» 


>- 


— X) XI 


— 


— 


v — ^ 


y 


o 


•*— - "2 n 


— 


T-t 


:-- * ca 


3 


L0 


xOh 


— 


S 


y — 3 


O 


_J 


— X 


T-l 


<i 


^-^ --~ 


o 


» 


vO — in *"* -* 


J_ 


• 


OCO.*H 


1 


( 


»-• O Tl C> •» 


< 


CVI 



l— ~ — Ul 



y 



x 



x 

X 



c 

LJ 



vO •» vi. •« H II » 

*- O — vi, S£ 

liiUUUiOH <V 

»— I- »■- *- X) ^ "S 

•— — >-••— V v 

X X X X CU 

< » Vs .i: d Z) 3 






56 



u 

X 

c 

r-t 

CD 

SD — 
_l V 

<- ^ 

* o 

• < 

O T 

as a 
H 11 



CO -3- 
V N 
13 13 



V 



no 



a. 

n 

H 3 
_ J V 

— 7 

If) •— »H 

■* •— >.n 

V 7" 



Z 

J- 
•— > 
X 

— CL 

3 o 

— 1-» 

»- a 

O _J 

■^-t < 

CD * 

<£ • 
_l o 
<: OJ 

•^-i ii ii 

+ «~ 
V *. 



in 






««» ^ 
o~ 
<■ "i 

X — 

— X 

33 

^ ^^ 
•*■• «-i 
M 3 

a. — 



CL 

X 
Cl 



cu 

V 



o 



o 



I 



OJ 



TO 
I 



> 
X 

3 

% — - 

D O 



o 



o 

CO 



ru 



cu 



o 



o 

I 



ru 



cu 



oc 



o 

cc 



ru 



ru 



^r 



r • 



7 



•* •— ■ CU U' 



II 



3 O C. 3 3 
O 

in 



U^ _•: 



ir 



«-• *-i CV1 t— *-i !* 

\£) u u ! J u 

u- •«-< CU L ' ! 

! _j i 

— >- > _i — X 

ll x x <t or x 

•S ~) "3 O jS J 



X 


X 


X 


X 


X 


>: 


3 


~i 


3 


«. — 


"» 


«— •_ 


3 OJ 


3 


— i " "j- 


~- — < 


• — 


■!—• ■ — 


<!— — » 


ro >— 


«-» in v— 


c? *» 


c 


«• 


_J >c 


l» 3 


sO II _l 


Ci. — 


G 


— • L^ 


> 


— ~> 


■"• Z> 


UJ 


OJ 


UJ (Vi 


_f •— 


— _J 


r _j 


• *— 


X _J 


— X 3 


•< ::' 


X < 


CC X < 


O X 


1U 


•- -) t 







3; 








ct 








Qt 








UL 








X 








»^ 








C3 








tu z 








3 ■« 








83 








en rr: 




-^. 




zo 




*• 




c u_ 




o 




CJ 




vO 




Ul 




4» 




Z rsj 




«* 




(B — 




o 




3 




vC 




»— <t 




II 




UJ «-*■ 




«• 




CO *— 




•c 




UJ •-• 




fU 




rr- ;£ 




««. 




•-*- 




m 




^_ 




ru 


Z 


UJ <D 




ii 


•— ► 


X h- 




^ 


< 


l — 




T-+ 


w 


"J~ 




^f 


< 


3 CJ 




»-T 




3 i— 


ry 


•f- 


h— 


r ■■ • , | 


^-f 


m 


-v 


►— i 3* 


m 


»-» 


<r 


CO 




•i- 


»— 


*T* 


e 


m 


CO 


CO -rH 


>— 


»-H 




3 CO 




^f 


o 


C_ O 


f; 


X 


h- 




C3 


X 




n i — 




~y 


?: 


3 U_ -^ 


«— 


*> 


UJ h— CO -d- 


O 


w* — 


3 


(.0 - ~ 


« 


3 d- 




►— 


i — 



U! 



— •— c tn '- -- lj 

OJ r- >h ir t- H a l • 

ct * a ot- r. 

i> 3 -C [C h 

Q — ■ v."> Ui — » —f '-' 

— > UJ Q f— U 

ru Ul 3 CC 3 OJ — <E 

— Jh^Dl C n — *— (1. 
X 3 — -~ I- _1 U 
x <i rr Z Z 3 Ll. U rr V- 
TUSliJwC >~^o w 



OJ 

»— I 
CO 



P0 



*■ *■ 



57 







>~ 


^~ 






















z: 






































CO 


CTk 






















en 






































CD 


OJ 






-^ 
















*—* 






































Is: 


— ' 






Lf) 
















H- 






































(— «-» 


CM 






^-« 
















a: 






































•^ 


CD 






^>' 
















CD 






































>- o> 


^ 






U 
















a. 






































co ~ 


— 






a. 






















































UJ Q (T> 






>- 
















CL 


































•-■ 




7 •». 


C\J 






^ 
















-) 


































CL 




CD — 


^~ 




»-» 


— 
















1 


































*. 




«» C\ 


CO 




CT^ 


in 
















J" 


































»— 




>- ai 


«t 




rvi 


■«-« 
















cr 


































^ 






* 






— 
















< 


































1- 




a u 


«— 




o 


VJ 
















3 


































_J 




cr * 


0\ 




< 


> 


















































UJ 




UJ *■» 


OJ 




_i. 


:d 








Ul 








-J^ 


































Q 




«v en 


1 — ' 




a 


^ 








■^ _ 








►— * 


































•» 




C3 CVJ 


•r-l 




* 


^^ 








• — t 










































*-i 




CO — 


d 




^ 


U) 








t— 








""> 


































CO 




-n ra 


^ 




cr« 


»-t 












!xJ 




OJ 


































7 




3 ^ 


"^ 


*-« 


a 


~~ 








a 


CJ 


r c^ 




CO 
























X 










cr; 




1- ~ 


cr. 


0> 


^ 


u 


•-» 






I - ) 


m 


^- OJ 










^^ 


— 
















CD 






•~« 


'-- 


LJ 


CD 


«» cr 


cv 


CV 


LJ > 


IT 






i 




t- 




c; 






V 


V 
















»— • 






V 


V 


^ 


3 


C CVI 


^ 


■■— 


> 


<t 


»-i 






; 


o ■ 


3 




H- 






»— 


— 
















h-- 






«— 


-~- 


x-l 


>— 


L" — 


-.-i 


LJ 


-* 


"» 


«— 






L". 


»— 


2 t— 










VJ 


CJ 
















tt 






LJ 


L. 


Q 


v 


IJ <C 


< 


U. 


.<— 


^^ 


T' 






*i 




r> 


— «. 


(t- 






* 


* 
















CTi 






* 


A 


•V 


LiJ 


li «\ 


^ 


«L 


i 


u: 


2: 






3S 


oc 


Cii CD 


cr 


^_-- 






•~> 


"^ 






«— i 






— 




a^ 






•^ 


■~ 


«- 1 


X 


CD ~- 


~-~ 


-^ 


O! 


T-* 


• — 4 








CO 


13 


i— 


~> 






V 


V 






LJ 






5 




^^ 






v 


M 


O 


CD 


«. C 


CT 


O 


^ 


^- 


JL 






CG 




OT 


7> 








^^ 


^— 






* 






^.- 




Z cu 






<" 


"^ 


«. 


«. 


O CO 


Ck. 


cv 


Q. 


a 


a 


*- * 




»— 1 


—~ 


•— < «•-'. 


r 

IK 


C: 






m 


<t 






*■» 






U! 




3 ^ 






CO 


<L 


h- 


UJ 


C/} — 


— 


w 


«C 


> 


^ 


_J 






CJ 


O 


» 


t— 














V 






ri/ 




a k 










• — • 


«■ 


CC 3 


UJ 


Q_ 


~ 




-^ 


a 




CO o 


o ri 


"O 








-f 


1 






^^ 






? 




~D 






+ 


1 


CC Zi 


:r 


cr -. 


a 


u 


0. 


•* 


Lf 


> 


•— i 


c 


a 


O L r , 


^ 


H 














Q 




•^-. 


— 




c/) a 










\- V 


t- 


..u *"* 


•» 


v. 


^ 


«-« 


■r-< 


< 


+ 


1 ' 


» ■ 


^i «-< 


I — 


-?_ 






^^ 


»-« 






> 




C"^ 


LJ 




•— • *« 






■^ 


— 


~^L •> 


"D 


* c 


■^ 


^ 


-— 


i: 


— 


> 


H- 


; 


* 


cr • 


7) 


• — ' 






V 


V 






<t 




y 


* 




CD 






V 


V 


— ♦ >— 


*j— 


O OJ 


c\ 


ETi 


cr 


T-l 


r» 


. 


^ 


U h 


LJ ♦— 


.'. 


^p. 




0^ 


~-~ 


--- 










Z~i 


•— • 




.1 ^ 




cr 


— 


«.» 


>' 


» 


co — 


n. 


CV! 




«— 


a 


2 




_L 


LI 


Uj cj 


•> 


r 




n 


* 


C -' 






+ 




LJ 


_1 




C U' 




a 


o 




LJ 13 


»— 


»- a 


— 


>^ 


^-- 


a 


•— • 


UJ 


C 


f o 


m 


C/3 • 


«— t 






^ 


* 


A 










* 


o 




1—1 — 




«. 


* 


3|E 


.£. CD 


.3 


. 




U 


Q 


> 


3 




\ 




1— 


(- 


•>— 


m 


c 


*— i 


■ — 


— 


»- 


»-i 


»— 




1- 


J._ 




H - 


o 


»-i 


^^ 


•— 


*— « * 


a. 


a < 




'-^ 


> 


< 


a 


j' 




o 




-i . ' 


UJ 


»— i 


• 


it 


V 


V- 


L 1 






:.. 


^ 


< 


*0 


- 


• 


»i 


V 


V 


r- 


















u 


•. • 


__.' 


.->"; .^ 


< 


_u 


vJ 


V 


--' 


^^ 






.' L. 


' "^ 






m 


i) 1 


v ) 


.V 




»- 


"~> Z 


7T 


2 7 


7" 


■*-* 


s 


^ 


^ 


s 






c 


^^ CD 


n 


►- 






< 


a. 


ii 


ii 




J 




11 




a_ <r 






«.r 


>A 


CD ii ;■ 


C. 


CD UJ 


CD 


. 


C 


<E 


d; 


••. 


h- 


»— 


V 


biT 






it 


O 










II 


•— . 


n 




CD 




11 


»— ( 






... 2 


4- 


y_ >. 


12 


z~ 


- 


1 




T* 


2 


:o 


— 


h- — 


_j 


r" 




t-i 


II 


ii 


«— 


— 




i — 




a 


»— 


CO _J 




CV! 


ii 


D 


f 


~ 


> 5 


5 


y 


t 


3 


N 


%" 


T* 


I- ' 




X 


_j 


XT 


H- 








V 


V 


h- 


_ 


»- 


»~ 




•— • 1 


»— 








lb x 


C 




a 


cr 


c 


CD 


c 


z:- 




t— 


u. 


Lu a 


< 


;;r 


ID 


CD 


T-» 


«-< 


•^ 


— 


*5 


. 


u 


~;. 


*• 


X <i 


D 


CT 


t— i 


T-» 


<j l. 


LJ 


l> l 


l. 


L 


<_' 


O 


t-^ 


U 


y 


u 


»— « 


» 


lj 


u 


a 


O 


L.' 


c 


u a 


Q 


U 


C' 


Q 


C? 


I- l; 


a 


L_: 


LJ 


, 














































o 








o 






















































tH 








^i 











58 






or 

UJ 

5>T 



(E> 






CO 
UJ 

^ <• 

u »- 

* I! 

-^ I 

V UJ 



2.. f- 

+ O £3 U «J — O 

Z> O U-. Cj <^ LJ T. s.- UJ 

a * ~z ?z * iz ^ ui 

h uj a? »— ui (L ■ t- a t 

»— z < 

~0 +-•-+ + + + <r-l — CY 

Q. co u to 

^- ~ ~ — ~ ~ ^:* 6D 

— :*: >c < m si \: cd •— :t. 

*~t UJ *-• fU UJ «-• CVJ * CZ 

u o u .j Zj < ro ►— --♦ 

_, II H H K II II J ■ C f! Ui 

w ii • u 00 ii r 

v^ --» — — -^ ^~ ^~ ►_. m £; fv; 

■^ — — a' v v .v v v <r- : t t— •— \— ZJ 

y v ^ ~ — -- ~ ~ ; »-- »— —. .— ►- o 

l ^j -■ ^ :j »-i w c z) z> j" ZJ Z) ui z 

u o •-* <. <t <t a l3 a ua i c v v e uj 



cz. 


ctf-- 


cr 


oc 


•r4- 


ur. •• 


M-Ul 


at+j- 


»d; 


cr^rr 


a u 


UL>4- 


at 


r-4-4-«- 


at.cr. 


4J-r+ 


•r+ 


GrcT 


•H-ar. 


fdrcu 


n 5 


a-p 


-M- QX 


nxr. 


Or 


a; a 


r+& 


(tt.-r+ 


ff-P 


KT- 


54- 


Ul OJ 


r-4 -M 


rH-^f: 


«r. CU 


ae 


a 


ai a 


cr 


•h- ar 


4J- > 


at 54 


a <o. 


w w 


A" ct 


3- o 


ta u 


CD o 


C:+J- 


-H 


5&<d 


0. m 


r4 54 


l-l a 


a >■ 


44 


(1JL 


ci a 


,Or4 


E4-> 



59 



2* 
G- 

Z> 
»- 

If) 



o 7 

_j — ■ t- »- 

^. «-» 5. O *H < 1J .3 Oi C 

Lu fc* Li; O O * tJ LJ \h C 



t. en 
a _i 

U ■: U 

Lj U Z «H 

if) < C >-" ' 



_i o 
u o 



o 

c 
o 
o c 

o _ 






4-uoitH n 



3 

. 

U> 



< >- 

JDDDtur uj uj Ll : uj uj ar 3 < < -r < < or D d co q < z \j ^ r :r h- x •: 3 < a • 

: C; C ;'N[i nnk \\ V Q C H »- ' "■ *~ v . - ;■ ' _J Q r- v .- •— iD CT . ' \ 3 O <T . 

j u» Li-- uj g. ^~ a o n a a w a _j to w _i c^ t;; . _j < ."• r no co lj a _» cr> ca _i u t 



< 



. 



< 






60 



o 

c 
o 

o «-• 

»— «. r^ •>-* — ■ 3l •»-< 

Z c_ < I s * x < it: •» <^ 

13 DOCN ^ sUWQIW X < 

< Q C »- : " < > < (.j -- - t L. + < U :.' 

* < CJ •— * ■ *— *- £j #» <£ O 4ft o o * 



< Do ~ 

< 

IT I 



o 
o 
o 
r- 

r-- 



■C *r* CD 7" T-' 

2" «« _l CD O 

cj •— u < lj < o -■ 

^ cu rv C-o <l 2 .'3 < u ;f> o 

. -^ o o <»-•-« i' ir r. < r" < 



^ O H Cz Ch Q 



*. 



II 



c o < u. 



>- >- o o >->->- < < 

q >- a x < oic < x n: tsj ^ h- cc a ; r > in a. a a. -/; ar o < x isi < c- n <: 2; »— ft: y 

»- v or l: ft* g e h ft or r~ ;r a c c or r- c c_ c ft- 1— o 1— or »- o _i n 1- c a 1 a 

o ; co v _i : 1 > c.j :/> c a en □ : r: _j u < u u <_> < u.' < co co _j _j < co ui a 



61 






o o 

a o 

•*> «k 

o o 

o o 

o o 

o c 

o o 

o o 

o o 

o c 

o o 

o o 

U. C O 

moo 



Lf 



*-* < o a 
«. *■»■«. r. 
<r. o o o o" 



; < <: 

LJ 3i I! CJ Ll! U* aJ Cl '— . Z •— K- CO ^l_. CO 3 

^ V rr :-" r- rvj r^ J r ._ .r <-. - l - . ll' I' 

a t/> as 32 a. a a u_ g. lj £"• r- cr„ c a uj 



o 



2T 


-J 


2 


U U 


iaj 


LJ LJ 


m t • 1 


' 


. 


CJ O 




'. -i 


u. c 


L 


L 


< < 


-^•. 





62 



LIST OF REFERENCES 



McRuer, D. T. and Krendel, E. S., "Dynamic Response of 
Human Operators," WADC-TR-56-524 , Wright-Patterson 
Air Force Base, Ohio, October, 1957. 

Krendel, E. S. and McRuer, D. T., "A Servomechanisms 
Approach to Skill Development," Journal of Franklin 
Institute , v. 269, No. 1, January, 1960. 

Elkind, J. I. and Green, D.M., "Measurement of Time- 
Varying and Nonlinear Dynamic Characteristics of 
Human Pilots." ASD Technical Report 61-225, Wright- 
Patterson Air Force Base, Ohio, February, 1962. 

Thomas, J. B., Statistical Communication Theory , Wiley, 
New York, 1969. 

Wylie, C. R. , Jr., Advanced Engineering Mathematics , 
McGraw-Hill, New York, 1960. 

Hess, R. A., "An Introduction to Human Describing 
Function and Remnant Measurement in Single Loop 
Tracking Tasks," AFFDL/FGC-TM-72-9 , Wright-Patterson 
Air Force Base, Ohio, May, 1972. 

Lee, Y. W. , Statistical Theory of Communication , Wiley, 
New York, 1960. 

McRuer, D. T. and Graham, D. , "Human Pilot Dynamics in 
Compensatory Systems," AFFDL-TR-65-15 , Wright- 
Patterson Air Force Base, Ohio, July, 1965. 



63 



HUTiaL. DISTRIBUTION hist: 

Nd^ . Copies 

1. Defense Documentation- Centex" 22 
Cameron Station 

Alexandria , Virginia 223T4 

2. Library, Code 0212 22 
Naval Postgraduate School 

Monterey , California 9194 

3. Asst. Professor R„ A.. Hess 11 
Department of Aeronautical Engineering 

Naval Postgraduate School 
Monterey , California 9 3 9 40 

4. Asst. Professor M. EL. Redlin II 
Department of Aeronautical Engineering 

Naval Postgraduate School 
Monterey, California 93 940 

5. LCDR Roy D. Warren, USN 11 
USS Ticanderoga ,. CVS-14 

%FPO 

San Francisco, California 9660X 

6 . Chairman 11 
Department of Aeronautics 

Naval Postgraduate School 
Monterey, California 93940 



64 



Socuntv Classification 



DOCUMENT CONTROL DATA -R&D 

{Security classification ol title, body ol abstract and indexing annotation roust be entered when the overall report Is classified) 



I ORIGINATING ACTIVITY (Corporate author) 

Naval Postgraduate School 

Mnnt-prpy, P^lifnrni^ 9^940 



¥ 



2«. REPORT SECURITY CLASSIFICATION 

Unclassified 



26. GROUP 



3 REPORT Tl TL 



A Hybrid Computer Technique for Measuring Human Describing Functions 
and Remnant in Closed-Loop Tracking Tasks 



4. DESCRIPTIVE NOTES (Type ol report and, inclusive dates\ 

Engineer's Thesis; June 1972 



9. AUTHORISI (First name, middle initial, last name) 



Roy D. Warren 



6 REPORT DATE 



June 1972 



7a. TOTAL NO. OF PAGES 



76. NO. OF REFS 



8a. CONTRACT OR GRANT NO. 



6. PROJEC T NO. 



9a. ORIGINATOR'S REPORT NUMBER(S) 



96. OTHER REPORT NO(S) (Any other numbers that may be assigned 
this report) 



10 DISTRIBUTION STATEMENT 



Approved for public release; distribution unlimited. 



II. SUPPLEMENTARY NOTES 



13. ABSTRACT 



12. SPONSORING Ml LI TAR Y ACTIVITY 



Naval Postgraduate School 
Monterey, California 93940 



The measurement of the human describing function and remnant 
in a compensatory tracking task is undertaken. These measurements 
are obtained through the application of the fast Fourier transform 
technique on a hybrid (analog-digital) computer. This method 
processes the data in real time with minimal core storage and the 
results are available immediately upon completion of the tracking 
run. 



DD, F , 



"" 1473 

1 NOV 63 • *T / <J 

S/N 0101 -807-681 1 



(PAGE 1) 



65 



Security Classification 



i-31408 



UNCLASSIFIED 



Security Classification 



KE<f WORDS 



LINK C 
I 



1. Human Describing Function 

2. Human Response Function 

3. Human Transfer Function 

4. Pilot Response 

5. Describing Function 

6 . Human Remnant 

7 . Remnant 



D , F r. M .,1473 IBACK, 



0101-907-68?! 



66 



UNCLASSIFIED 



Security Classification 



» - 3 I -109 



Thesis 135519 

W22968 Warren 

c.l A hybrid computer 

technique for measur- 
ing human describing 
functions and remnant 
in closed-loop track- 
ing tasks. 





Thesis 
W22968 
c.l 



135519 



Warren 

A hybrid computer 
technique for measur- 
ing human describing 
functions and remnant 
in closed-loop track- 
ing tasks. 



thesW22968 




3 2768 001 92976 ^ 



■ 
I